The concept of illegal immigration is a modern one: it requires a state to legislate immigration. As such, illegal immigration is a product of modern statehood. Seen in this way, European colonists were not illegal immigrants. There were no laws banning them from coming to the Americas.
In the United States, immigration was relatively unrestricted until the late 1800s–early 1900s. American immigration policy became more restrictive over time, with ebbs and flows. See the first link below.
To provide a more nuanced answer to the question, one would need to explore Native American perceptions of the colonists. The colonists were not illegal immigrants, but were the settlers unwanted guests? How did native perceptions vary from tribe to tribe, region to region, and over time? There is an extensive corpus of scholarly literature on this topic (see the second link below for an example); however, owing to the cultural gap between natives and Europeans and the paucity of extant native sources, many questions remain unanswered.